Found problems: 85335
VI Soros Olympiad 1999 - 2000 (Russia), 8.8
Let $p_1$, $p_2$, $...$, $p_n$ be different prime numbers ($n\ge 2$). All possible products containing an even number of coefficients (all coefficients are different) are composed of these numbers. Let $S_n$ be the sum of all such products. For example, $$S_4 = p_1p_2 + p_1p_3 + p_1p_4 + p_2p_3 + p_2p_4 + p_3p_4+ p_1p_2p_3p_4.$$ Prove that $S_n + 1$ is divisible by $2^{n-2}$.
2015 Peru Cono Sur TST, P8
Let $ABCD$ be a cyclic quadrilateral such that the lines $AB$ and $CD$ intersects in $K$, let $M$ and $N$ be the midpoints of $AC$ and $CK$ respectively. Find the possible value(s) of $\angle ADC$ if the quadrilateral $MBND$ is cyclic.
2021 CMIMC, 1.6
Alice and Bob each flip $20$ fair coins. Given that Alice flipped at least as many heads as Bob, what is the expected number of heads that Alice flipped?
[i]Proposed by Adam Bertelli[/i]
1975 Swedish Mathematical Competition, 4
$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.
2003 China Team Selection Test, 1
$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.
2018 Romanian Master of Mathematics Shortlist, G1
Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$.
[i]Italy[/i]
1993 Irish Math Olympiad, 1
The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$
2004 Unirea, 3
[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality
$$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$
holds.
[b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?
2015 IMAR Test, 3
Let $ABC$ be a triangle, let $A_1, B_1, C_1$ be the antipodes of the vertices $A, B, C$, respectively, in the circle $ABC$, and let $X$ be a point in the plane $ABC$, collinear with no two vertices of the triangle $ABC$. The line through $B$, perpendicular to the line $XB$, and the line through $C$, perpendicular to the line $XC$, meet at $A_2$, the points $B_2$ and $C_2$ are defined similarly. Show that the lines $A_1A_2, B_1B_2$ and $C_1C_2$ are concurrent.
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.
2010 Purple Comet Problems, 21
Let $a$ be the sum of the numbers:
$99 \times 0.9$
$999 \times 0.9$
$9999 \times 0.9$
$\vdots$
$999\cdots 9 \times 0.9$
where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$.
2001 Federal Math Competition of S&M, Problem 2
Given are $5$ segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.
1994 National High School Mathematics League, 3
Sequence $(a_n)$ satisfies that $3a_{n+1}+a_n=4(n\geq1),a_1=9$, let $S_n=\sum_{i=1}^{n}a_i$, then the minumum value of $n$ such that $|S_n-n-6|<\frac{1}{125}$ is
$\text{(A)}5\qquad\text{(B)}6\qquad\text{(C)}7\qquad\text{(D)}8$
1992 China Team Selection Test, 3
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
2004 China Team Selection Test, 3
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.
2012 Kosovo National Mathematical Olympiad, 2
In a sphere $S_0$ we radius $r$ a cube $K_0$ has been inscribed. Then in the cube $K_0$ another sphere $S_1$ has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.
2007 ISI B.Math Entrance Exam, 9
In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .
2017 India PRMO, 23
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.
1984 Tournament Of Towns, (066) A5
Let $p(n)$ be the number of partitions of the natural number $n$ into natural summands. The diversity of a partition is by definition the number of different summands in it. Denote by $q(n)$ the sum of the diversities of all the $p(n) $ partitions of $n$.
(For example, $p(4) = 5$, the five distinct partitions of $4$ being $4, 3 + 1, 2+2, 2 + 1 + 1, 1 + 1 + 1 + 1,$ and $g(4) =1 + 2+1+ 2+1 = 7$.)
Prove that, for all natural numbers $n$,
(a) $q(n)= 1 + P(1) + P(2) + p(3) + ...+ p(n -1)$,
(b) $q(n) < \sqrt{2n} p(n)$.
(AV Zelevinskiy, Moscow)
2015 India IMO Training Camp, 3
Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.
2002 Korea - Final Round, 3
Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$
(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying
\[2\le r \le n-2, \quad n-r+1 < p_r\]
and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$.
(b) Using the result of (a), find all positive integers $m$ for which
\[p_{m+1}^2 < p_1p_2\cdots p_m\]
1986 USAMO, 4
Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.
Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.
2015 CCA Math Bonanza, L2.1
What is the sum of the first $10$ primes?
[i]2015 CCA Math Bonanza Lightning Round #2.1[/i]
2009 Purple Comet Problems, 13
Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$
2009 BAMO, 3
There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^2$ and $b^2$ end in the same last two digits. For example, $35^2 = 1225$ and $45^2 = 2025$ both end in $25$. What are all possible values for the average of $a$ and $b$?
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^2$ to end with the digits 04, not $4$.